Problem 22 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (2 x+3)^{5} $$

Step-by-Step Solution

Answer

\( (2x+3)^5 = 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243 \)

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] . Here, \( a = 2x \), \( b = 3 \), and \( n = 5 \).

Step 2: Identify the terms for the formula

Using the values identified: \( a = 2x \), \( b = 3 \), and \( n = 5 \) , we will expand \( (2x + 3)^5 \).

Step 3: Calculate the binomial coefficients

Calculate \( \binom{5}{k} \) for all values of \( k \) from 0 to 5. These are the coefficients for each term in the expansion: \[ \binom{5}{0} = 1 \quad \binom{5}{1} = 5 \quad \binom{5}{2} = 10 \quad \binom{5}{3} = 10 \quad \binom{5}{4} = 5 \quad \binom{5}{5} = 1 \] .

Step 4: Apply the Binomial Theorem

Plug in the values into the Binomial Theorem: \[ (2x + 3)^5 = \binom{5}{0} (2x)^{5} (3)^{0} + \binom{5}{1} (2x)^{4} (3)^{1} + \binom{5}{2} (2x)^{3} (3)^{2} + \binom{5}{3} (2x)^{2} (3)^{3} + \binom{5}{4} (2x)^{1} (3)^{4} + \binom{5}{5} (2x)^{0} (3)^{5} \] .

Step 5: Simplify each term

Simplify each term step by step: \[ (2x + 3)^5 = 1 \cdot (2x)^{5} \cdot 1 + 5 \cdot (2x)^{4} \cdot 3 + 10 \cdot (2x)^{3} \cdot 9 + 10 \cdot (2x)^{2} \cdot 27 + 5 \cdot (2x)^{1} \cdot 81 + 1 \cdot (2x)^{0} \cdot 243 \] .

Step 6: Finalize the calculations

Perform the final multiplications and powers for each term: \[ (2x + 3)^5 = 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243 \] .

Key Concepts

Binomial Theorem

Binomial Theorem is a fundamental principle used to expand expressions that are raised to a power. It states that any binomial expression \[ (a + b)^n \] can be expanded into a sum involving terms of the form \[ \binom{n}{k} a^{n-k} b^k \]. The binomial theorem allows us to write out the polynomial expansion without directly multiplying the terms repeatedly. In our example, we expanded \[ (2x + 3)^5 \] using the binomial theorem.
To break it down:

• \[ a \] and \[ b \] are the terms inside the binomial (here, \[ 2x \] and \[ 3 \]).
• \[ n \] is the power to which the binomial is raised (in this case, 5). This exponent tells us the number of terms in the expansion.

Utilizing the binomial theorem can simplify complex polynomial expansions and reduces the chances of errors. Each term in the expanded form represents a binomial coefficient multiplied by the necessary powers of the two terms being added together.

Binomial Coefficients

Binomial Coefficients are the numbers that appear in the binomial theorem expansion. They are represented as \[ \binom{n}{k} \] and are calculated using the formula:

• \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

The factorial notation (e.g., \[ n! \]) means multiplying a series of descending natural numbers. For instance, 5! (read 'five factorial') is \[ 5 \times 4 \times 3 \times 2 \times 1 \].
In our example, for \[ (2x + 3)^5 \], the binomial coefficients are \[ \binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \] and \[ \binom{5}{5} \]. Each coefficient multiplies the corresponding terms in the binomial expansion.
These coefficients are crucial since they determine the weight and contribution of each term in the expanded form. They also ensure the balance and symmetry in the polynomial expressions.

Polynomial Expansion

Polynomial Expansion involves breaking down a polynomial expression into a sum of simpler polynomials. When we expand \[ (2x + 3)^5 \] using the binomial theorem, we get a sum of several simpler polynomials:

• \[ (2x + 3)^5 = 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243 \]

Each term is a product of powers of \[ 2x \] and 3, weighted by the binomial coefficients. This process makes it easier to understand and work with complex polynomial expressions by breaking them into more manageable parts. Polynomial expansion finds applications in algebra, calculus, and even in fields like cryptography and coding theory.

Exponentiation

Exponentiation refers to the operation of raising a number or expression to a given power. It is a fundamental concept in mathematics, especially in algebra. In our binomial theorem example, we dealt with exponentiating the terms \[ 2x \] and 3 in each phrase of the expansion. For instance:

• \[ (2x)^5 = 32x^5 \]
• \[ (3)^1 = 3 \]
• \[ (3)^2 = 9 \]

Knowing how to handle exponentiation correctly is essential for using the binomial theorem effectively. It helps in multiplying terms precisely and consolidating the final expanded form of the expression.
In the final result for our example, each term is formed by multiplying the binomial coefficient, the exponential powers of the first term \[ 2x \], and the exponential powers of the second term (constant) 3. Mastery of exponentiation helps with accuracy and simplifies polynomial and algebraic solutions.